Extrude with Proportional Editing
Not directly, but extrude works like this: when you press E, blender 


creates new connected geometry
immediately enters "grab" mode (but not using Proportional)


If you cancel(ESC or right click), you cancel only the last "grab" action, thus leaving the new geometry where it was before entering the "grab mode", and still selected.

You could issue this as a bug or a feature request, though, imho.

So, you can just enter "grab mode" again (eg pressing G) but now your enabled Proportional Editing will work